The Standard Model

1. The Standard Model Thomas J. LeCompte High Energy Physics DivisionArgonne National Laboratory

2. The Direction of These Lectures • I’m an experimenter • I will start with simple models, and build up to the Standard Model • Models will be progressively more predictive, but… • This isn’t strictly chronological • I will touch on QCD only lightly • Themes • What data needs a new model to explain it? • Where might the Standard Model be wrong? • And how could we tell? I’ll (over)emphasize tests that might be performed at the LHC.

3. Part I: Quantum Electrodynamics A Theory with Just One Parameter

4. Local Gauge Invariance – Part I • In quantum mechanics, the probability density is the square of the wavefunction: P(x) = |Y|2 • If I change Y to –Y, anything I can observe remains unchanged • P(x) = |Y|2 can be perhaps better written as P(x) = YY* • If I change Y to Yeif anything I can observe still remains unchanged. • The above example was a special case (f = p) • If I can’t actually observe f, how do I know that it’s the same everywhere? • I should allow f to be a function, f(x,t). • This looks harmless, but is actually an extremely powerful constraint on the kinds of theories one can write down.

5. Local Gauge Invariance – Part II • The trouble comes about because the Schrödinger equation (and its descendants) involves derivatives, and a derivative of a product has extra terms. • At the end of the day, I can’t have any leftover f’s – they all have to cancel. (They are, by construction, supposed to be unobservable) • If I want to write down the Hamiltonian that describes two electrically charged particles, I need to add one new piece to get rid of the f’s: a massless photon.

6. Local Gauge Invariance & QED • Add a phase to the electron wavefunction • The derivative brings a phase out front, but… • There has to be an extra term in the covariant derivative representing the photon field • And the photon field has to transform properly.

7. Why Massless? • A massive spin-1 particle has three spin states (m = 1,0,-1) • A massless spin-1 particle has only two. • Hand-wavy argument: Massless particles move at the speed of light; you can’t boost to a frame where the spin points in another direction. • To cancel all the f’s, I need just the two m = ± 1 states (“degrees of freedom”) • Adding the third state overdoes it and messes up the cancellations • The photon that I add must be massless m = ±1 “transverse” m = 0 “longitudinal” Aside: this has to be just about the most confusing convention adopted since we decided that the current flows opposite to the direction of electron flow.We’re stuck with it now.

8. U(1), Lie Algebras and All That • The key abstraction is the same equations have the same solutions • “You’ve seen one U(1) theory, you’ve seen them all” • U(1) stands for “unitary group of degree one” • We used to call those things “numbers” • The U(1)-ness comes in at the qAm term • Here the charge is just a number • Instead of a number, we can make charge a matrix • e.g. QCD color: {red, blue, green} • Then the SU(3)-ness manifests as matrix multiplication: QaAam • SU(2) is the same group that governs the algebra of angular momentum • Why we have “spin” and “isospin” • Again, the same equations have the same solutions

9. Fermions – The Red-Headed Stepchildren of SM Talks • 12 massive spin-½ particles • 3 charged leptons • 3 neutral leptons • 6 colored quarks • Masses vary by 11-13 orders of magnitude • There are three families – nobody knows why • There is no evidence whatever that the families are as ordered in the left. (e.g. the tau could be the partner of the u & d quarks)

10. Why Are Atoms Neutral? • Why are atoms neutral? • Or, why is q(p) = -q(e) • Or, why do quarks and leptons have charges related by small integer ratios? • This is not an accident • One needs to avoid “anomalies” – quantum mechanical inconsistencies (e.g infinities) in the calculations • The troublesome terms have simple algebraic relations between the charges (e.g. Sq) as coefficients • Make the coefficients zero, and all is well.

11. QED • The Good: • Incredible precision: ge = 2.00231930436 (and every digit is significant) • The Bad: • Doesn’t explain nuclear b decay • Inconsistent at very high energy • So-called “Landau Pole” • Very high energy means E > m(visible universe) The next step: adding a theory of weak interactions that doesn’t spoil what has already been accomplished…

12. Part II: Fermi’s Four Fermion Theory A Theory with Two Parameters

13. Four Fermion Theory ne m • The idea is just like the name says • Four fermions couple directly • The coupling strength is GF e nm • I write it this way (rather than 292 GeV) because this number will be important later – and I want you to remember where it came from. • The seeds of the theory’s destruction are already planted …but let’s talk about the theory’s successes before talking about its failures.

14. Some successes • Nuclear b decay rates should go as ~Q5 • Observed in 1933 as “Sargent’s Rule”. • Explained by simple power counting • Any decay rate ~ width G ~ dimensions of energy • The 4-fermion interaction has 2 powers of GF (and thus four powers of energy) in the denominator • Therefore you need five powers of Q in the numerator • The same GF that explains nuclear b decays explains • m decay • p decay • n interactions & cross-sections

15. The Pion Decay: Evidence of handedness (chirality) p+ (Anti-) Lepton Neutrino • The pion is spin-0, so the daughters must have the same helicity • The decay rate depends on whether the interaction can couple to both left and right handed electrons, or only one Only one Both Measurement: 8100

16. Which Handedness? • If the weak interaction is left-handed (V-A) • The neutrino is left-handed • The anti-lepton has left-handed helicity • But the weak interaction needs to couple to the right-handed chiral projection • This costs us a ml/mp in the amplitude (squared in the rate) • Suppose the weak interaction were right-handed (V+A) • Now the neutrino is right-handed • The anti-lepton has right-handed helicity • But the weak interaction needs to couple to the left-handed chiral projection • This still costs us a ml/mp in the amplitude (again, squared) The pion branching fractions tell us that there is a handedness to the weak interaction. But it doesn’t tell us which hand it is.

17. Measuring the Neutrino Helicity M. Goldhaber, 1958 • K-capture on the right nucleus hasthe helicity of the daughter nucleusmatch the neutrino’s. • Next, the daughter emits a photon • Carries the spin of the parent • Transfers the neutrino spin to the photon The essential idea is that the neutrino’s helicity (hard to measure) is transferred to the photon (easy – well, easier – to measure). Outcome:the weak interaction is left handed.

18. Strangeness and the Cabibbo Angle • Kaon decays have all the same qualitative properties as other weak decays • Except they proceed ~30x slower • N. Cabibbo explained this by theorizing: • 1. The weak interaction only allowstransitions within a doublet • 2. The weak interaction eigenstates are rotated with respect to the mass eigenstates • The s-quark mass eigenstate contains only ~3% dW, so these decays are suppressed by this factor of 30. For the same reason – the charged lepton has the “wrong” helicity. Experiment agrees. Weak interactions are universal – the same force that governs nuclear b decay governs heavy flavor decays.

19. Kobiyashi & Maskawa: Extending this to Three Families Can be expressed in terms of three angles and one phase – the 9 terms are not independent Aside: the phase here gives rise to CP violation. Three is the minimum number of families for this to happen.

20. Cabibbo, Kobiyashi & Maskawa Matrix II Numbers don’t give me a very good intuition for what’s going on • Because the CKM matrix appears squared in any observable, it acts even more like a diagonal matrix • The weak interaction apparently does not like to cross family boundaries Here the shading reflects the magnitude of the components: black = 1 and white = 0.

21. Cross-sections • A cross-section has units of area: length2 (or energy-2) • Dimensional analysis tells us any exclusive cross-section must (eventually) fall as 1/s, i.e. 1/E2. What are we to make of the following data? (The cross-section for p-p scattering)The cross-section is actually growing with energy. Energy

22. Extending The Energy • The cross-section grows because it’s the left side of a resonance, the D(1232) • The high energy behavior of elastic scattering is OK: 1/E2 • The total high energy behavior is also OK – it’s the sum of many channels Energy

23. Lessons for the Weak Interaction • Any weak cross-section must grow as E2 and not 1/E2. • Any cross section has two powers of GF (1/Energy4), so it needs two additional powers of E to get the right units. • This doesn’t depend on any of the details of the calculation, or even on field theory • It’s simply dimensional analysis • That means that the theory will get into trouble – and violate unitarity – somewhere around 300 GeV.

24. Fermi Four-Fermion Theory Scorecard • The Good: • Explains b decay, weak decays and neutrino interactions in detail • Surprise – it’s a chiral theory • Left and right handed fermions behave differently • Parity is violated • The Bad: • Theory breaks down somewhere before ~300 GeV The next step: add new physics (the W’s and Z) to the theory to fix it up past 300 GeV

25. Part III: Glashow-Weinberg-Salam Theory A Theory with Three Parameters

26. A Troublesome Event m nm A charged current event – the outgoing muon has a different charge than the incoming neutrino. nm A picture of an event from the CERN Gargamelle bubble chamber. nm This incomprehensible plot from 1973 shows “neutral weak currents”. A neutral current event – the outgoing neutrino has the same charge as the incoming neutrino.

27. Some Experimental Issues The trick is to distinguish this from this. nm m nm nm “How long is long” becomes a key question. It’s the presence of a long muon track that distinguishes charged current from neutral current events. But what if the muon ends up with relatively little momentum? nm m A second issue is “How do you know these are neutrinos? Might they be neutrons?”That can be addressed by looking at where the interactions occur – are they uniform, like neutrinos? Or are they near a detector edge, like neutrons?

28. What The Authors Were Probably Thinking Of… • They weren’t trying to explain neutral weak currents • In fact, neutral weak currents were predicted years before their discovery. • They were trying to unify QED with a theory of weak interactions • Their theory predicted weak neutral currents and the Wand Z bosons

29. The Roadmap • We want QED to come out • So let’s start by putting it (or something very much like it) in • We want a left-handed weak theory to come out • So let’s start by putting it (or something very much like it) in • Let’s pick a small group that will let us have charged currents… • …and at least the possibility of neutral currents • Once we’ve done this, we will match terms to pull QED out • What’s left will be our (new) theory of weak interactions

30. First Ingredient: Weak Hypercharge • This is where the Lie Algebra formalism starts to help us: • “We start with a U(1)” • This means that we will end up with a theory just like QED • Instead of a field Am, we call it (unimaginatively) Bm • Instead of a charge q, we have a “hypercharge” Y. • Replace qAm with YBm in the Lagrangian and we’re done. The Same Equations Have The Same Solutions

31. Second Ingredient: Weak Isospin • In this case, our gauge group is SU(2) • This is the same algebra that governs angular momentum addition • It’s a non-Abelian group – the fields themselves carry weak isospin • We have three fields: w1, w2 and w3. • They interact among themselves in this way w1 w1 w2 w1 w3 w2 w2

32. Interaction Between these Fields and Matter Weak Hypercharge Weak Isospin • Here g1 and g2 are the coupling constants for hypercharge and isospin • Right now, they are arbitrary (the ½ is there by convention) • The j’s are the fermion currents • Our next step is to declare electric charge to be: Q = ½Y + T3. • Essentially, this specifies which w is neutral and which are charged • The absolute magnitude of charge is rolled up in the g’s • I haven’t really lost any generality by doing this

33. A Change of Basis We should now cast everything in terms of electric charge – we want QED to fall out of this: Then becomes Neutral Positive Negative

34. Charged Currents: What Exactly Have We Done? n m n m We’ve replaced the 4-fermion interaction with two vertices where the fermion current couples to a W field. W e e n n Now we have everything we need to calculate the same process in the old 4-fermion theory and the new GWS theory. If we do this, and match the results, we get: We just calculated this vertex factor: it’s m W n

35. Neutral Currents We have no guarantee that the w3 and B are the physical fields (in fact, they aren’t) so we introduce a rotation matrix to mix them. (in the new basis)

36. Neutral Currents We have no guarantee that the w3 and B are the physical fields (in fact, they aren’t) so we introduce a rotation matrix to mix them. (in the new basis) But we already know from QED what this has to be! So we simply match up terms…

37. More On Neutral Currents • Matching up terms gives us a relationbetween the weak and EM couplings • They are NOT INDEPENDENT • I can plug this into the Z term, and get By matching terms, we effectively calculated this vertex factor: it’s Note that the weak coupling is larger than the EM coupling: aweak ~ 1/30 vs. 1/137 for aEM. The weak force is weak not because the coupling is small, but because the W is heavy. f Z f

38. Enough Math…Time For Some Physics • The magnitude of the axial Z coupling to fermions is universal • The photon is a vector, so it doesn’t touch the axial piece of the weak interaction • The more electric charge a fermion has, the less (vector) weak charge it has • The photon is “stealing” the charge

39. Some Predictions This evaluates to 77.5 GeV. The measurement is 80.4 GeV. Using the measured W mass, this evaluates to 91.6 GeV.The measurement is 91.2 GeV. All these predictions use qW measured independently - from neutrino experiments.

40. The W Boson in Pictures From the original discovery (UA1/UA2)This is a UA2 event D0 CDF Missing ET (neutrino) Electron momentum

41. The Z Boson In Pictures OPAL UA1 DELPHI CDF ALEPH L3

42. g W+ W+ Additional Consequences of SU(2) We originally introduced SU(2) because we wanted the W to be charged (“charged current”). That means it has to couple to the photon: This is the physical manifestation of the 3-field interaction in the unbroken SU(2) theory. The physical states are projections of w1, w2 and w3. w3 w1 w2 Z0 But these aren’t the only projections one can make.The theory predicts the W couples to the Z as well. W+ W+

43. W+Z Events CDF D0 Not only does this theory predict the existence of these events, the rate of these events is completely determined. Z muon W electron More on this tomorrow. Z muon W neutrino

44. A Word on Symmetries • This theory is often described as SU(2) x U(1) • Since electromagnetism also has a U(1) theory, it’s tempting to associate EM with the above U(1) • This is wrong • The unbroken symmetry is SU(2)Left x U(1)Hypercherge • We explicitly broke both symmetries when we declared the w3 component to be electrically neutral • We will discuss another way to break these symmetries tomorrow • Often, a calculation that looks terrible in the broken symmetry is simple if you consider the unbroken symmetry basis. If you discipline yourself to always write SU(2)L x U(1)Y, you’ll avoid making this mistake.

45. Fixing the Unitarity Problem… • The Fermi theory has a problem with unitarity violation above 300 GeV • There must be new physics below that. • We added new physics below that: the W and Z’s at 80 and 90 GeV • We’ve fixed that problem • The cross-sections drop like they are supposed to. Compare This is actually a picture of what happens in the neutral currents, and I originally argued this for charged currents. The same thing happens for both– this is simply a clearer plot.

46. Fixing the Unitarity Problem…Almost • Imagine you could collide beams of W bosons: WW → WW • If you calculate this in GWS theory, it too violates unitarity • Specifically, one piece: WLWL → WLWL • New physics has to enter in below about 1 TeV • So we haven’t so much as solved the problem as moved it to higher energy. Potential Troublemaker

47. GWS Theory Scorecard • The Good: • Matches every test against data we could think of • Predicts new phenomena, borne out by experiment • W and Z bosons • Neutral Weak Currents • Diboson production at colliders • Explains everything with just three numbers • GF, the strength of the weak force, a, the strength of the EM force, and qw, how they mix • Fixes the 300 GeV Unitarity problem of the Fermi Theory • The Bad: • Theory breaks down above ~1 TeV • Masses put in by hand • Breaks gauge invariance • Symmetry broken by hand Our next step – fixing these problemsby adding one more piece to the theory.